Journal of Applied Mathematics and Physics, 2013, 1, 49-53
Published Online November 2013 (http://www.scirp.org/journal/jamp)
Open Access JAMP
Exact Solution and Conservation Laws for Fifth-Order
Korteweg-de Vr ies Equation
Elham M. Al-Ali
Mathematics Department, Faculty of Science, University of Tabuk, Tabuk, KSA
Received September 11, 2013; revised October 11, 2013; accepted October 17, 2013
Copyright © 2013 Elham M. Al-Ali et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using
the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order
KdV equation is considered.
Keywords: Soliton Solutions; Conservation Laws; Nonlinear Evolution Equations
It is well-known that nonlinear complex physical phe-
nomena are related to nonlinear partial differential equa-
tions (NLPDEs) which are involved in many ﬁelds from
physics to biology, chemistry, mechanics, etc. As mathe-
matical models of the phenomena, the investigation of
exact solutions to the NLPDEs reveals to be very important
for the understanding of these physical problems. Many
mathematicians and physicists have well understood this
importance when the importance of this so they decided to
pay special attention to the development of sophisticated
methods for constructing exact solutions to the NLPDEs.
Thus, a number of powerful methods have been presented.
We can cite the inverse scattering transform , the
Bäcklund and Darboux transform [2-5], Hirota’s bilinear
method , the homogeneous balance method , Jacobi
elliptic function method , the tanh-method and ex-
tended tanh-function method [9-15], F-expansion method
[16-18] and so on.
The notion of conservation laws is important in the
study of nonlinear evolution equations (NLEEs) appear-
ing in mathematical physics . The mathematical ori-
gin of conservation laws results from the formulation of
familiar physical laws such as for mass, energy and mo-
mentum . As is known, the investigation of conser-
vation laws of the Korteweg-de Vries (KdV) equation led
to the discovery of a number of techniques to solve
NLEEs , e.g., Miura transformation, Lax pair, in-
verse scattering technique and bi-Hamiltonian structures.
On the other hand, it is useful in the numerical integra-
tion of NLEEs  (e.g., to control numerical errors);
particularly with regard to integrability and linearization,
constants of motion, analysis of solutions, and numerical
solution methods . Consider a dynamical system,
,uuxt is a function of two independent
variables t and x. The functional
is said to
be a constant of motion or an integral of Equation (1), if
Generally, we can derive
constants of motion from conservation law, which enjoys
the general form as 
Vu xtG u xt
while the components V and of the conserved
,VG are functions of ,
tand derivatives of
. The equality (2) is assumed to be satisfied for any
solution of the corresponding system of equations, is
called conserved density and G is called conserved
flow. With the assumption that the function
its derivatives with respect to
go to zero sufficiently
fast as x
is obtained to be a constant of motion. It has already been
proved that a large number of NLEEs possess an infinite
E. M. AL-ALI
number of conservation laws such as the fifth-order KdV
2. Exact Solution for Fifth-Order KdV
With the rapid development of science and technology,
the study kernel of modern science is changed from lin-
ear to nonlinear step by step. Many nonlinear science
problems can simply and exactly be described by using
the mathematical model of nonlinear equation. Up to
now, many important physical nonlinear evolution equa-
tions are found, such as sin-Gordon equation, KdV equa-
tions, Schrodinger equation all possess solitary wave
solutions. There exist many methods to seek for the soli-
tary wave solutions, such as inverse scattering method,
Hopf-Cole transformation, Miura transformations, Dar-
boux transformation and Bäcklund transformation [2-5],
but solving nonlinear equations is still an important task.
In this paper, with the aid of Mathematica, a traveling
wave solution for a class of fifth-order KdV equation
uu uuuuuu x
In order to obtain the soliton solution of (4), the soli-
tary wave ansatz is assumed as
is the soliton amplitude, is the width of
the soliton, is the soliton velocity and
to be determined later, the unknown index n will be de-
termined during the course of derivation of the solution
of Equation (4). From Equation (5), I obtain Equation (6).
With the aid of Mathematica or Maple, from (5) and
uating the exponents
), we can get Equation (7).
Now, from Equation (7) eq
leads 1332nn, which gives
From (7) setting the coeffici
zero, I get Equations (8)-(10).
sech sinh41 sech sinh
4 1sechsinh6 12sechsinh
An nAn nn
sech sinh1 sech sinh
ABd nAd nABd nAd nABd n
Ad nAd nAd nAd nAd n
Open Access JAMP
E. M. AL-ALI 51
Solving the above system by the aid of Wu elimination
method , I obtain the two solutions
304016 0,Bc Bdd
224Bd AB Add
Then the soliton solutions of the fifth order KdV equa-
is given by tion
3. Systematic Construction Method
nfinitely Many Conservation L
ifth-Order KdV Equation
We recall the definition [16,23] of a differe
aws for I
) that describes a pss. Let 2
be a two dimensional
differentiable manifold with coordinates
t. A DE
for a real function a nec-
essary and sufficient conditifor the exi
depending on u and its derivatives such that the
pss if it is a
stence of dif-
isfy the structure equations of a pss, i.e.
As a consequence, each solution of the Dovides a
local metric on
DE for is the integrability
condition for the problem [14,26]:
se Gaussian curvature is con-
stant, equal to −1. Moreover, the above definition is
equivalent to saying that u
where denotes exterior differentiation,
is a col-
umr and the 2 × 2 matrix
ij ,,ij is
from Equations (18) and (19), we obtain
xAt qxBtSx Tt
where S and T are two 2 × 2 null-trace matrices
is a paeter, independent of ram
while q and r are functions of
and t. Now
0d dΩΩddΩΩ ,Ω
which requires the vanishing of the two form
or in component form
rC Ar C
22 12 3212 32
11 3111 31
Chern and Tenenblat  obtain
rectly from the structure equations (17). By suitably
choosing and in (24), we shall obtain vari-
ous fifthV uation wh
Konno and Wadati introduced the function 
ed Equation (24) di-
eqich q must satisfy.
this function first appeared used and explained in the
[11,13], and see also the classical pap
27]. Then Equati
geometric context of pseudo spherical eq
ers by Sasaki 
on (20) is reduced and Chern-Tenenblat [
to the Riccati equations
Equations (28) and (29) imply that
Open Access JAMP
E. M. AL-ALI
to both sides and using the expression
qrfrom (24), Equation (30) takerm s the fo
let us show how an infinite number of conservatio
result from these results. The Riccati equations for in
variable can be rearranged to take the form
A similar pair of equations can be obtained for the t
derivatives. Expand Гr into a power series in the
are unknown at this point, however a recursion
relation can be obtained for the n
by using (32), sub-
stituting (33) into the Г equation in (32), I find that
rq x t
Appling the Cauchy product formul
in (34), then I obtain
rq x t
Now equate powers of
on both sides of this ex-
pression to produce the set of recursions,
Substituting (33) into (31), the followin
conservation laws appears
g system of
This procedure generates an infinite number of con-
servation laws for the equation under examination. To
obtain conservation laws using (37) in a particular exam-
ple using this procedure, let us consider the fifth-order
KdV Equation (4), for Equation (4)
43 2 2
cu u uuu
qruAu u uu
nto (24), I obtain the fifth-order
tting (38) into (37), it is found that
Substituting (38) i
KdV Equation (4). Pu
ear wave that
possesses remarkable stability properties. Typically, prob-
lems that admit soliton solutions are in the form o
lution equations that describe how some variable or set of
variables evolve in time from a given state. The equa-
tions, partial difference equa-
tions, and integro-differential equations, as well as cou-
pled ODEs of finite order.
In this paper, we considered the construction of exact
hysics and applied
mathematics. Solitons are found in various areas of
physics from hydrodynamics and plasma physics, non-
linear optics and solid state physics, to field theory and
gravitation. NLEEs which describe soliton ph
have a universal character.
A travelling wave of permanent form has already been
met; this is the solitary wave solution of the NLEE itself.
Such a wave is a special solution of the governing equa-
The Soliton equations play a central role in the field of
integrable systems and also play a fundamental role in
several other areas of mathematics and physics.
and P. A. Clarkson, “Solitons, Nonlinear
tion and Inverse Scattering,” Cambridge
A soliton is a localized pulse-like nonlin
ns may take a variety of forms, for example, PDEs,
differential difference equa
lutions to fifth-order KdV equation. We obtain travelling
wave solutions for the above equation by using the soli-
tary wave ansatz method with the aid of Mathematica.
The soliton phenomena and conservation laws of
NLEEs represent an important and well established field
of modern physics, mathematical p
n which does not change its shape and which propa-
gates at constant speed.
 M. J. Abolwitz
University Press, Cambridge, 1991.
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E. M. AL-ALI
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 C. Rogers and P. Wong, “On reciprocal B€ a Acklund
Transformations of Inverse Scattering Schemes,” Physica
Scripta, Vol. 30, 1984, pp. 10-14.
 A. H. Khater, D. K. Callebaut, A. A. Abdalla, A. R. She-
hata and S. M. Sayed, “Backlund Transformations and
Exact Solutions for Self-Dual SU(3) Yang-Mills Equa-
tions,” IL Nuovo Cimento B, Vol. 114, 1999, pp. 1-10.
 C. Qu, Y. Si and R. Liu, “On Affine Sawada-Kotera
Equation,” Chaos, Solitons & Fractals, Vol. 15, No. 1,
2003, pp. 131-139.
 O. C. Wright, “The Darboux Transformation of Some
Manakov Systems,” Applied Mathematics Letters, Vol.
16, No. 5, 2003, pp. 647-652.
 R. Hirota, “The Direct Method in Soliton Theory,” Cam-
bridge University Press, Cambridge, 2004.
 A. H. Khater, D. K. Callebaut and S. M. Sayed, “Exact
Solutions for Some Nonlinear Evolution Equations which
Describe Pseudospherical Surfaces,” Journal of Computa-
tional and Applied Mathematics, Vol. 189, No. 1-2, 2006,
pp. 387-411. http://dx.doi.org/10.1016/j.cam.2005.10.007
 S. K. Liu, Z. T. Fu and S. D. Liu, “Jacobi Elliptic Func-
tion Expansion Method and Periodic Wave Solutions of
Nonlinear Wave Equations,” Physics Letters A, Vol. 289,
No. 1-2, 2001, pp. 69-74.
 E. Fan, “Extended Tanh-Function Method and Its Appli-
cations to Nonlinear Equations,” Physics Letters A, Vol
277, 2000, pp. 212-219.
 W. Malfliet and W. Hereman, “The Tanh Method I. Exact
Solutions of Nonlinear Wave Equations,” Physica Scripta,
Vol. 54, No. 6, 1996, pp. 569-575.
K. Chadan and P. C. Sabatier, “Inverse Problem in Quan-
tum Scattering Theory,” Springer, New York, 1977.
 M. J. Ablowitz, S. Chakravarty and R. Hal
Painlevé and Darboux-Halphen Type Equatio
ns, in the
Painlevé Property, One Century Later,” In: R. Conte, Ed.,
CRM Series in Mathematical Physics, Springer, Berlin,
M. Elham Al-Ali, “Traveling Wave Solutions for a Ge n-
eralized Kawahara and Hunter-Saxton Equations,” Inter-
national Journal of Mathematical Analysis, Vol. 7, 2013,
S. M. Sayed, “The Bäcklund Transformations, Exa ct So-
lutions, and Conservation Laws for the Compound Modi-
fied Korteweg-de Vries-Sine-Gordon Equations which
describe Pseudospherical Surfaces,” Journal of Applied
Mathematics, Vol. 2013, 2013, pp. 1-7.
 V. B. Matveev and M. A. Salle, “Darboux Transforma-
tions and Solitons,” Springer-Verlag, Berlin, 1991.
 K. Tenenblat, “Transformations of Manifolds and Appli-
cations to Deferential Equations, Pitman Monographs and
Surveys in Pure and Applied Mathematics 93,” Addison
Wesley Longman, England, 1998.
 A. M. Wazwaz, “New Compactons, Solitons and Periodic
Solutions for Nonlinear Va
Equations,” Chaos, Solitons & Fractals, Vol. 22, 20
riants of the KdV and the KP
 A. M. Wazwaz, “Two Reliable Methods for Solving Vari-
ants of the KdV Equation with
Structures,” Chaos, Solitons & Fractals, Vol. 28, No.
Compact and Noncompact
2006, pp. 454-462.
 X. G. Geng and H. Wang, “Coupled Ca
tions, N-Peakons and Infinitely Many Conservat
Journal of Mathematical Analysis and Applications, Vol.
403, 2013, pp. 262-271.
 A. H. Khater, D. K. Callebaut and S. M. Sayed, “Conser-
vation Laws for Some Nonlinear Evolution Equations
which Describe Pseudo-Spherical Surfaces,” Journal of
Geometry and Physics, Vol. 51, No. 3, 2004, pp. 332-352.
 J. A. Cavalcante and K. Tenenblat, “Conservation Laws
for Nonlinear Evolution Equations,” Journal of Mathe-
matical Physics, Vol. 29, 1988, pp. 1044-1059.
 R. Beals, M. Rabelo and K. Tenenblat, “Backlund Trans-
formations and Inverse Scattering Solutions for Some
Pseudo-Spherical Surfaces,” Studies in Applied Mathe-
matics, Vol. 81, 1989, pp. 125-134.
 E. G. Reyes, “Conservation Laws and C
Deformations of Equations Describing
Surfaces,” Journal of Mathematical Physics, Vol. 41, 2000,
pp. 2968-2979. http://dx.doi.org/10.1063/1.533284
 E. G. Reyes, “On Geometrically Integrable Equations and
Hierarchies of Pseudo-Spherical Type,” Contemporary
Mathematics, Vol. 285, 2001, pp. 145-156.
 W. T. Wu, “Polynomial Equations-Solving and Its Ap-
ons,” Studies in Applied Mathemat-
plications,” Algorithms and Computation, Beijing, 1994,
 M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,
“The Inverse Scattering Transform-Fourier Analysis f
Nonlinear Problems,” Studies in Applied Mathematics,
Vol. 53, 1974, pp. 249-257.
 S. S. Chern and K. Tenenblat, “Pseudospherical Surfaces
and Evolution Equati
ics, Vol. 74, 1986, pp. 55-83.
 K. Konno and M. Wadati, “Simple Derivation of Back-
lund Transformation from Riccati Form of Inverse Me-
thod,” Progress of Theoretical Physics, Vol. 53, 1975, pp.
erical Sur-  R. Sasaki, “Soliton Equations and Pseudosph
faces,” Nuc lear Physics B, Vol. 154, 1979, pp. 343-357.